The effective dynamics of interacting waves for coupled Schrödinger-Korteweg-de Vriesequations over a slowly varying random bottom is rigorously studied. One motivation forstudying such a system is better understanding the unidirectional motion of interactingsurface and internal waves for a fluid system that is formed of two immiscible layers. Itwas shown recently by Craig-Guyenne-Sulem [1] thatin the regime where the internal wave has a large amplitude and a long wavelength, thedynamics of the surface of the fluid is described by the Schrödinger equation, while thatof the internal wave is described by the Korteweg-de Vries equation. The purpose of thisletter is to show that in the presence of a slowly varying random bottom, the coupledwaves evolve adiabatically over a long time scale. The analysis covers the cases when thesurface wave is a stable bound state or a long-lived metastable state.

We study the spectral stability of solitary wave solutions to the nonlinear Diracequation in one dimension. We focus on the Dirac equation with cubic nonlinearity, knownas the Soler model in (1+1) dimensions and also as the massive Gross-Neveu model.Presented numerical computations of the spectrum of linearization at a solitary wave showthat the solitary waves are spectrally stable. We corroborate our results by findingexplicit expressions for several of the eigenfunctions. Some of the analytic results holdfor the nonlinear Dirac equation with generic nonlinearity.

We consider two atomic transitions excited by two variable laser fields in a three-level system. We study the soliton-pair propagation out of resonance and under thermal bath effect. We present general analytical implicit expression of the soliton-pair shape. Furthermore, we show that when the coupling to the environment exceeds a critical value, the soliton-pair propagation through three-level atomic system will be prohibited.

A linear, uniformly stratified ocean model is used to investigate propagation of baroclinic Kelvin waves in a cylindrical basin. It is found that smaller wave amplitudes are inherent to higher mode individual terms of the obtained solutions that are also evanescent away of a costal line toward the center of the circular basin. It is also shown that the individual terms if the obtained solutions can be visualized as spinning patterns in rotating stratified fluid confined in a circular basin. Moreover, the fluid patterns look rotating in an anticlockwise sense looking above the North Pole and that spinning is more intensive for smaller mode numbers. Finally, we observe the existence of the oceanic region where the pressure increases relatively rapidly with the depth.

Today engineering and science researchers routinely confront problems in mathematical modeling involving solutions techniques for differential equations. Sometimes these solutions can be obtained analytically by numerous traditional ad hoc methods appropriate for integrating particular types of equations. More often, however, the solutions cannot be obtained by these methods, in spite of the fact that, e.g. over 400 types of integrable second-order ordinary differential equations were summarized in voluminous catalogues. On the other hand, many mathematical models formulated in terms of nonlinear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating nonlinear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is, from the one hand, to provide the wide audience of researchers with the comprehensive introduction to Lie’s group analysis and, from the other hand, is to illustrate the advantages of application of Lie group analysis to group theoretical modeling of internal gravity waves in stratified fluids.

In this paper, we show finite time blow-up of solutions of the p−waveequation in ℝN, with critical Sobolev exponent. Our workextends a result by Galaktionov and Pohozaev [4]

Two-dimensional inviscid channel flow of an incompressible fluid is considered. It isshown that if the flow is steady and features no horizontal stagnation, then the flow mustnecessarily be a parallel shear flow.

The influence of a global delayed feedback control which acts on a system governed by asubcritical complex Ginzburg-Landau equation is considered. The method based on avariational principle is applied for the derivation of low-dimensional evolution models.In the framework of those models, one-pulse and two-pulse solutions are found, and theirlinear stability analysis is carried out. The application of the finite-dimensional modelallows to reveal the existence of chaotic oscillatory regimes and regimes withdouble-period and quadruple-period oscillations. The diagram of regimes resembles thosefound in the damped-driven nonlinear Schrödinger equation. The obtained results arecompared with the results of direct numerical simulations of the original problem.

We review some recent results concerning Gibbs measures for nonlinear Schrödingerequations (NLS), with implications for the theory of the NLS, including stability andtypicality of solitary wave structures. In particular, we discuss the Gibbs measures ofthe discrete NLS in three dimensions, where there is a striking phase transition tosoliton-like behavior.

A simple explicit numerical scheme is proposed for the solution of the Gardner–Ostrovskyequation (ut + cux + α uux + α1u2ux + βuxxx)x = γuwhich is also known as the extended rotation-modified Korteweg–de Vries(KdV) equation. This equation is used for the description of internal oceanic wavesaffected by Earth’ rotation. Particular versions of this equation with zero some ofcoefficients, α, α1, β, orγ are also known in numerous applications. The proposed numericalscheme is a further development of the well-known finite-difference scheme earlier usedfor the solution of the KdV equation. The scheme is of the second order accuracy both ontemporal and spatial variables. The stability analysis of the scheme is presented forinfinitesimal perturbations. The conditions for the calculations with the appropriateaccuracy have been found. Examples of calculations with the periodic boundary conditionsare presented to illustrate the robustness of the proposed scheme.

The matrix KdV equation with a negative dispersion term is considered in the right upperquarter–plane. The evolution law is derived for the Weyl function of a correspondingauxiliary linear system. Using the low energy asymptotics of the Weyl functions, theunboundedness of solutions is obtained for some classes of the initial–boundaryconditions.

We obtain solvability conditions in H6(ℝ3) for asixth order partial differential equation which is the linearized Cahn-Hilliard problemusing the results derived for a Schrödinger type operator without Fredholm property in ourpreceding article [18].